arXiv subjects

Juha Kontinen

Publications and source records attributed to Juha Kontinen.

At least 19 recordsLinked to original sources

arXiv2026

Inquisitive first-order logic is neither compact nor recursively axiomatizable

Inquisitive first-order logic is an extension of classical first-order logic with formulas regimenting first-order questions, such as "whether all objects are P", "which objects are P", and "what is one object that is P". Since it was first developed in 2009, two major meta-theoretical questions about this logic have remained open, in spite of significant efforts. The first concerns compactness: if a conclusion follows from a set of premises, does it always follow from some finite subset? The second concerns the computational status of validity: is the set of validities recursively enumerable, or equivalently, does the logic admit a recursive axiomatization? We settle both questions in the negative, showing that inquisitive first-order logic is neither compact nor recursively axiomatizable. Furthermore, we prove that it violates another signature property of first-order logic, namely, Craig interpolation. We discuss the significance of our results, and show how to extend them to a closely related logic, viz., inquisitive team logic.

math.LO↗

arXiv2026

Intuitionistic Implication in Elementary Team Logics

The logic FOT is a team-based logic whose expressive power coincides with first-order logic at the level of both sentences and open formulas. In contrast to dependence and independence logics, which can define stronger second-order team properties, FOT is designed to capture exactly elementary team properties, modulo the empty team. In this paper we consider two modifications of FOT. First, we investigate essentially the inclusion atom free fragment of FOT. Our main result establishes quantifier elimination for the fragment in the empty signature. Second, we study an extension of FOT by the intuitionistic implication. The main conclusion is that adding this single connective increases the expressive strength so that every second-order sentence can be encoded by an open formula evaluated on the full team. Consequently, validity of formulas is equivalent to validity of full second-order logic.

math.LO↗

arXiv2026

On the Expressive Power of Inquisitive Team Logic and Inquisitive First-Order Logic

Inquisitive team logic is a variant of inquisitive logic interpreted in team semantics, which has been argued to provide a natural setting for the regimentation of dependence claims. With respect to sentences, this logic is known to be expressively equivalent with first-order logic. In this article we show that, on the contrary, the expressive power of open formulas in this logic properly exceeds that of first-order logic. On the way to this result, we show that if inquisitive team logic is extended with the range-generating universal quantifier adopted in dependence logic, the resulting logic can express finiteness; as a consequence, this logic is not compact and has non-arithmetic complexity. We further extend our results to standard inquisitive first-order logic, showing that some sentences of this logic express non first-order properties of models, thus settling an open problem from the literature.

math.LO↗

arXiv2026

Aspects of Coherence in Dependence Logic

Dependence logic extends first-order logic with dependence atoms asserting that the value of a variable is determined by the values of certain other variables. The semantics of dependence logic has a second-order character and involves sets of assignments, called teams, instead of individual assignments as in the classical Tarski semantics. Since the model-checking problem is known to be NP-complete even for quantifier-free dependence logic (DQF) formulas, researchers have pursued conditions on formulas that make this problem tractable. In 2010, Jarmo Kontinen introduced the notion of k-coherence for dependence logic formulas, where k is a positive integer. This notion asserts that if the formula is satisfied in a structure by all k-element subteams of a given team, then the given team itself satisfies the formula. It has been proved that k-coherent DQF-formulas have a tame model-checking problem, because such formulas admit a first-order rewriting. In this paper, we investigate the structural and algorithmic aspects of coherence. We show that if a DQF-formula is first-order ewritable, then it is k-coherent for some positive integer k. Thus, for DQF-formulas, coherence is equivalent to first-order rewritability. Furthermore, we show that an analogous result holds for universally quantified dependence logic formulas under a stronger notion of coherence. After this, we focus on the complexity of deciding if a given dependence logic formula is k-coherent. We establish that this decision problem is highly undecidable for arbitrary dependence logic formulas, while for DQF-formulas this problem is co-recursively enumerable. Furthermore, we pinpoint the computational complexity of the coherence problem for propositional dependence logic formulas by showing that this problem is complete for the second level of the exponential hierarchy.

cs.LO↗

arXiv2026

On the Complexity of Entailment for Cumulative Propositional Dependence Logics

This paper establishes and proves complexity results for entailment for cumulative propositional dependence logic and for cumulative propositional logic with team semantics. As recently shown, cumulative logics are famously characterised by System~C and exactly captured by the cumulative models of Kraus, Lehmann and Magidor. This gives rise to the entailment problem via relational models, which is specifically considered here.

cs.LO↗

arXiv2026

Complexity Results in Team Semantics: Nonemptiness Is Not So Complex

We initiate the study of the complexity-theoretic properties of convex logics in team semantics. We focus on the extension of classical propositional logic with the nonemptiness atom NE, a logic known to be both convex and union closed. We show that the satisfiability problem for this logic is NP-complete, that its validity problem is coNP-complete, and that its model-checking problem is in P.

cs.LO↗

arXiv2026

Representation Theorems for Cumulative Propositional Dependence Logics

This paper establishes and proves representation theorems for cumulative propositional dependence logic and for cumulative propositional logic with team semantics. Cumulative logics are famously given by System C. For propositional dependence logic, we show that System C entailments are exactly captured by cumulative models from Kraus, Lehmann and Magidor. On the other hand, we show that entailment in cumulative propositional logics with team semantics is exactly captured by cumulative and asymmetric models. For the latter, we also obtain equivalence with cumulative logics based on propositional logic with classical semantics. The proofs will be useful for proving representation theorems for other cumulative logics without negation and material implication.

cs.LO↗

arXiv2026

Locally Consistent K-relations: Entailment and Axioms of Functional Dependence

Local consistency arises in diverse areas, including Bayesian statistics, relational databases, and quantum foundations, and so does the notion of functional dependence. We adopt a general approach to study logical inference in a setting that enables both global inconsistency and local consistency. Our approach builds upon pairwise consistent families of K-relations, i.e, relations with tuples annotated with elements of some positive commutative monoid. The framework covers, e.g., families of probability distributions arising from quantum experiments and their possibilistic counterparts. As a first step, we investigate the entailment problem for functional dependencies (FDs) in this setting. Notably, the transitivity rule for FDs is no longer sound, but can be replaced by two novel axiom schemas. We provide a complete axiomatisation for, and establish NL-completeness of, the entailment problem of unary FDs, and demonstrate that even this restricted case exhibits context-dependent subtleties. In addition, we explore when contextual families over the Booleans have realisations as contextual families over various monoids.

quant-ph↗

arXiv2025

Independence Under Incomplete Information

We initiate an investigation how the fundamental concept of independence can be represented effectively in the presence of incomplete information in relational databases. The concepts of possible and certain independence are proposed, and first results regarding the axiomatisability and computational complexity of implication problems associated with these concepts are established. In addition, several results for the data and the combined complexity of model checking are presented. The findings help reduce computational overheads associated with the processing of updates and answering of queries.

cs.DB↗

arXiv2025

Logical Approaches to Non-deterministic Polynomial Time over Semirings

We provide a logical characterization of non-deterministic polynomial time defined by BSS machines over semirings via existential second-order logic interpreted in the semiring semantics developed by Grädel and Tannen. Furthermore, we show that, similarly to the classical setting, the satisfiability problem of propositional logic in the semiring semantics is the canonical complete problem for this version of NP. Eventually, we prove that the true existential first-order theory of the semiring is a complete problem for the so-called Boolean part of this version of NP.

cs.LO↗

arXiv2025

Towards New Characterizations of Small Circuit Classes via Discrete Ordinary Differential Equations

Implicit computational complexity is a lively area of theoretical computer science, which aims to provide machine-independent characterizations of relevant complexity classes. % for uniformity with subsequent uses >> 1960s (but feel free to modify it) % One of the seminal works in this field appeared in the 1960s, when Cobham introduced a function algebra closed under bounded recursion on notation to capture polynomial time computable functions ($FP$). Later on, several complexity classes have been characterized using \emph{limited} recursion schemas. In this context, an original approach has been recently introduced, showing that ordinary differential equations (ODEs) offer a natural tool for algorithmic design and providing a characterization of $FP$ by a new ODE-schema. In the present paper we generalize this approach by presenting original ODE-characterizations for the small circuit classes $AC^0$ and $FTC^0$.

cs.CC↗

arXiv2025

Characterizing Small Circuit Classes from FAC^0 to FAC^1 via Discrete Ordinary Differential Equations

In this paper, we provide a uniform framework for investigating small circuit classes and bounds through the lens of ordinary differential equations (ODEs). Following an approach recently introduced to capture the class of polynomial-time computable functions via ODE-based recursion schemas and later applied to the context of functions computed by unbounded fan-in circuits of constant depth (FAC^0), we study multiple relevant small circuit classes. In particular, we show that natural restrictions on linearity and derivation along functions with specific growth rate correspond to kinds of functions that can be proved to be in various classes, ranging from FAC^0 to FAC^1. This reveals an intriguing link between constraints over linear-length ODEs and circuit computation, providing new tools to tackle the complex challenge of establishing bounds for classes in the circuit hierarchies and possibly enhancing our understanding of the role of counters in this setting. Additionally, we establish several completeness results, in particular obtaining the first ODE-based characterizations for the classes of functions computable in constant depth with unbounded fan-in and Mod 2 gates (FACC[2]) and in logarithmic depth with bounded fan-in Boolean gates (FNC1).

cs.CC↗

arXiv2025

The flattening operator in team-based logics

We propose a systematic study of the so-called flattening operator in team semantics. This operator was first introduced by Hodges in 1997, and has not been studied in more detail since. We begin a systematic study of the expressive power this operator adds to the most well-known team-based logics, such as dependence logic, anonymity logic, inclusion logic and exclusion logic.

cs.LO↗

arXiv2025

Logic and Computation through the Lens of Semirings

We study the expressivity and computational aspects of first-order logic and its extensions in the semiring semantics developed by Grädel and Tannen. We characterize the complexity of model checking and data complexity of first-order logic both in terms of a generalization of Blum-Shub-Smale machines and arithmetic circuits defined over a semiring. In particular, we give a logical characterization of constant-depth arithmetic circuits by an extension of first-order logic that holds for any semiring that is both commutative and positive.

cs.LO↗

arXiv2025

On the Complexity and Properties of Preferential Propositional Dependence Logic

This paper considers the complexity and properties of KLM-style preferential reasoning in the setting of propositional logic with team semantics and dependence atoms, also known as propositional dependence logic. Preferential team-based reasoning is shown to be cumulative, yet violates System~P. We give intuitive conditions that fully characterise those cases where preferential propositional dependence logic satisfies System~P. We show that these characterisations do, surprisingly, not carry over to preferential team-based propositional logic. Furthermore, we show how classical entailment and dependence logic entailment can be expressed in terms of non-trivial preferential models. Finally, we present the complexity of preferential team-based reasoning for two natural representations. This includes novel complexity results for classical (non-team-based) preferential reasoning.

cs.AI↗

arXiv2024

A Remark on the Expressivity of Asynchronous TeamLTL and HyperLTL

Linear temporal logic (LTL) is used in system verification to write formal specifications for reactive systems. However, some relevant properties, e.g. non-inference in information flow security, cannot be expressed in LTL. A class of such properties that has recently received ample attention is known as hyperproperties. There are two major streams in the research regarding capturing hyperproperties, namely hyperlogics, which extend LTL with trace quantifiers (HyperLTL), and logics that employ team semantics, extending truth to sets of traces. In this article we explore the relation between asynchronous LTL under set-based team semantics (TeamLTL) and HyperLTL. In particular we consider the extensions of TeamLTL with the Boolean disjunction and a fragment of the extension of TeamLTL with the Boolean negation, where the negation cannot occur in the left-hand side of the Until-operator or within the Global-operator. We show that TeamLTL extended with the Boolean disjunction is equi-expressive with the positive Boolean closure of HyperLTL restricted to one universal quantifier, while the left-downward closed fragment of TeamLTL extended with the Boolean negation is expressively equivalent with the Boolean closure of HyperLTL restricted to one universal quantifier.

cs.LO↗

arXiv2024

Logics with probabilistic team semantics and the Boolean negation

We study the expressivity and the complexity of various logics in probabilistic team semantics with the Boolean negation. In particular, we study the extension of probabilistic independence logic with the Boolean negation, and a recently introduced logic FOPT. We give a comprehensive picture of the relative expressivity of these logics together with the most studied logics in probabilistic team semantics setting, as well as relating their expressivity to a numerical variant of second-order logic. In addition, we introduce novel entropy atoms and show that the extension of first-order logic by entropy atoms subsumes probabilistic independence logic. Finally, we obtain some results on the complexity of model checking, validity, and satisfiability of our logics.

cs.LO↗

arXiv2024

A Primer for Preferential Non-Monotonic Propositional Team Logics

This paper considers KLM-style preferential non-monotonic reasoning in the setting of propositional team semantics. We show that team-based propositional logics naturally give rise to cumulative non-monotonic entailment relations. Motivated by the non-classical interpretation of disjunction in team semantics, we give a precise characterization for preferential models for propositional dependence logic satisfying all of System P postulates. Furthermore, we show how classical entailment and dependence logic entailment can be expressed in terms of non-trivial preferential models.

cs.AI↗